$\begin{array}{l} x \in \left[ {\dfrac{1}{4};1} \right]\\ \Rightarrow \sin \pi x \in \left[ {\sin \pi ;\sin \dfrac{\pi }{4}} \right]\\ \Rightarrow \sin \pi x \in \left[ {0;1} \right]\\ \Rightarrow 0 \le \sin \pi x \le 1\\ \Rightarrow \left\{ \begin{array}{l} \min \sin \pi x = 0 \Rightarrow x = 1\\ \max \sin \pi x = \sin \dfrac{\pi} 2=1 \Rightarrow x = \dfrac{1}{2} \end{array} \right. \end{array}$